Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 10 + \left(8 a + 4\right)\cdot 23 + 15 a\cdot 23^{2} + \left(6 a + 1\right)\cdot 23^{3} + \left(2 a + 1\right)\cdot 23^{4} + a\cdot 23^{5} + \left(9 a + 3\right)\cdot 23^{6} + \left(10 a + 17\right)\cdot 23^{7} + \left(17 a + 10\right)\cdot 23^{8} +O\left(23^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 17 + \left(15 a + 4\right)\cdot 23 + \left(20 a + 10\right)\cdot 23^{2} + \left(17 a + 15\right)\cdot 23^{3} + \left(22 a + 20\right)\cdot 23^{4} + 21 a\cdot 23^{5} + \left(18 a + 15\right)\cdot 23^{6} + \left(a + 7\right)\cdot 23^{7} + \left(9 a + 3\right)\cdot 23^{8} +O\left(23^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 14 + \left(14 a + 18\right)\cdot 23 + \left(7 a + 22\right)\cdot 23^{2} + \left(16 a + 21\right)\cdot 23^{3} + \left(20 a + 21\right)\cdot 23^{4} + \left(21 a + 22\right)\cdot 23^{5} + \left(13 a + 19\right)\cdot 23^{6} + \left(12 a + 5\right)\cdot 23^{7} + \left(5 a + 12\right)\cdot 23^{8} +O\left(23^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 6\cdot 23 + 13\cdot 23^{2} + 21\cdot 23^{3} + 14\cdot 23^{4} + 22\cdot 23^{5} + 14\cdot 23^{6} + 4\cdot 23^{7} +O\left(23^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 7 + \left(7 a + 18\right)\cdot 23 + \left(2 a + 12\right)\cdot 23^{2} + \left(5 a + 7\right)\cdot 23^{3} + 2\cdot 23^{4} + \left(a + 22\right)\cdot 23^{5} + \left(4 a + 7\right)\cdot 23^{6} + \left(21 a + 15\right)\cdot 23^{7} + \left(13 a + 19\right)\cdot 23^{8} +O\left(23^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 + 16\cdot 23 + 9\cdot 23^{2} + 23^{3} + 8\cdot 23^{4} + 8\cdot 23^{6} + 18\cdot 23^{7} + 22\cdot 23^{8} +O\left(23^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)$ |
| $(1,2)(3,5)$ |
| $(1,2,4)(3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(2,5)$ |
$-1$ |
| $6$ |
$2$ |
$(2,4)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,4)(3,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,3,2)$ |
$-1$ |
| $6$ |
$4$ |
$(1,3)(2,6,5,4)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,6,3,2,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
